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1answer
49 views

Unable to validate the Roe matrix for the Shallow Water Equations

In LeVeque's Finite Volume Methods for Hyperbolic Problems, p. 320-321, one may find the derivation of the Roe matrix to the 1D Shallow Water Equations (SWEs). It is $$ ...
2
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0answers
60 views

Finite Volume/difference scheme for 2d continuity equation

I am trying to solve the following 2D continuity equation on the rectangular domain [0,1]x[0,1] using the finite volume method. $\rho_t + \nabla \cdot (\rho v) = 0$ where the velocity $v =(v1,v2)$ is ...
0
votes
1answer
66 views

Space discretization in Finite Volume Methods

Consider the linear system $$ u_t + Au_x = 0, $$ Discretize this using the explicit Euler method with constant mesh width $\Delta t$ in time and a finite volume method with constant mesh width ...
2
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1answer
93 views

Finite volume method implementation issues

I am trying to write a simple finite volume method code but there are some concepts I'm still not really getting right (perhaps I'm overcomplicating things) Given a uniform grid, the idea is to ...
1
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1answer
58 views

Finite Volume Method flux integration

I was reading through this document about FVM. I understood all up to the point where we have on page 15 the following $$(\bar u^{n+1}_i - \bar u^n_i) \Delta x + \int^{t^{n+1}}_{t^n} ...
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1answer
103 views

What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as $$ \left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( ...
2
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0answers
55 views

Euler Equation Eigensystem with Gravity in the Energy Flux

I am modifying a conservative form of the Euler equations with gravity in the energy flux (see previous question: Energy Conservation in Conservation Laws with Source Terms) for use in a Riemann ...
2
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1answer
139 views

Energy Conservation in Conservation Laws with Source Terms

I'm wondering if anyone can help me understand energy conservation when using conservation law methods (i.e. Riemann solver, High-Resolution Wave-Propagation Methods) with the addition of source ...
3
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1answer
94 views

Numerically solve a PDE in Python with a term calculated by coarse-graining

I'm trying to solve a PDE in Python of the form, $\dfrac{\partial c(\mathbf{x}, t)}{\partial t} = \mathrm{D} \nabla^2 c(\mathbf{x}, t) -\gamma \rho(\mathbf{x}, t) c(\mathbf{x}, t)$ where $c$ ...
0
votes
1answer
93 views

a circular plot from a vector which represents the temperature along the radius surface, which is the same for every radius

I have calculated the temperature of the section of a cylinder, which is subjected to a heat flow on its upper surface. Getting the temperature distribution in the 2D section. As shown in the ...
1
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1answer
54 views

Accurate dot-product of fields with only knowing normals

I am trying to accurately calculate $\vec{j} \cdot \vec{E}$ for an electron energy equation on a finite-volume mesh. $\vec{j}$ is the electron current density, and $\vec{E}$ is the electric field. ...
2
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2answers
112 views

Is there a bound on the number of edges, facets, and elements in a 3D simplicial mesh in terms of the number of mesh nodes?

I am currently working on a 2D finite element code with a mesh that contains duplicate nodes at certain interfaces where jumps occur. In order to set up the appropriate linear systems, I have to take ...
2
votes
2answers
104 views

Finite element: handle discontinuity/abrupt interface

I am using Galerkin's method to set up some pde solver for my problems(1D/2D) at hand. Some abrupt material interfaces do exist, so far, what I did regarding this is simply: 1) no element across the ...
1
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0answers
82 views

boundary conditions with non-constant coefficients in cell centred finite volume method

Suppose am solving the heat conduction equation in 1d with Dirichlet boundary conditions. The thermal conductivity $k$ is a non constant function. So $-(k(x)u'(x))' = f(x)$ The value of $k$ enters ...
3
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1answer
245 views

Comparison of Lattice Boltzmann Method vs Traditional Navier-Stokes based Methods

I have a choice of two options, analysing and implementing Lattice Boltzmann methods or traditional Navier Stokes based methods. I'm a CFD newbie and I have a rough idea (though not rigorous enough to ...
1
vote
1answer
165 views

Moving airfoil boundary conditions

I am trying to simulate a moving airfoil with constant speed (Mach=0.755, aoa=1.25). I solve Euler equations with Roe's method. I have two boundary conditions: Farfield Slip wall (airfoil) For all ...
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0answers
116 views

finite volume for diffusion equation with anisotropic (tensor) coefficient

Consider the scalar PDE for $u$ with Dirichlet boundary conditions: $\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$, $u = 0 \; \forall \; x\;\in \partial\Omega$ ...
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0answers
40 views

Interface Formulation at Finite Volume Boundaries when using the Dual Mesh

When using the dual mesh (vertex-centered) for finite volume methods, you end up with a cell center at the boundaries between materials. It is possible that the equations being solved in each ...
1
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0answers
77 views

data structures for efficient/easy implementation of finite volume method for 2D Poisson equation

My question is about implementation alone. Consider a square domain with regular square, cell centred finite volumes. This is for the multiscale finite volume method (Jenny and Lunati) I need to ...
1
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0answers
103 views

Assembling sparse matrix in PETSC for Poisson equation

I am a novice at PETSC, and I have been trying to write an FVM code for steady heat conduction in 2D using PETSC (square, regular grid, Dirichlet boundaries) Since the large matrix , say A, will be ...
3
votes
2answers
561 views

Developing finite volume (FVM) code in C . General advice

I need to develop a FVM code in C (The multiscale FVM method for heterogeneous media). I know that: Only uniform rectangular grids will be considered (2d now, later 3d) Sparse systems will be large ...
1
vote
1answer
199 views

Euler's equations 1d for pipe, Inlet boundary conditions

$\def\rmin{{\mathrm{in}}}$ $\def\l{\left}\def\r{\right}$ $\def\tagl#1{\tag{#1}\label{#1}}$ I am using the one-dimensional finite volume method to calculate the air flow in some tube. For subsonic ...
3
votes
1answer
67 views

Conservative FV Immersed boundary method for compressible flow

Is there a conservative FV second-order (or first-order) accurate immersed boundary method for compressible flow including moving boundaries (in the literature)? By compressible flow I mean the ...
6
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0answers
88 views

Implementation of convection scheme given by normalized variable diagram

In finite difference and finite volume methods, convection schemes (upwind, central, quick, ...) are usually shown in a normalized variable diagram. The diagram gives the normalized face variable as a ...
4
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0answers
111 views

Finite volume method

I have question connected with finite volume method. Consider equation $$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f . \quad x\in \overline{B}_{1} (0)\subset \mathbb{R}^3 -\text{unit ...
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2answers
331 views

Reconstructing fluxes [closed]

Given a standard advection equation, we write the update as $$ q_i^{n+1}=q_i^n+\frac{\Delta t}{\Delta x}\left(F_i^{n+1/2}-F_{i+1}^{n+1/2}\right) $$ with $F_i^{n+1/2}=F\left(q_i^n,\,q_i^*\right)$ and ...
5
votes
1answer
178 views

Finite Volume Implementation

I am trying to implement a simple finite volume method solver. I had a class on FVM a while back, but am still aware of the principal concepts. But implementing the FVM for non-cartesian or 1D meshes ...
1
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0answers
66 views

Characteristic length of differential element of cylinder surface?

I am trying to find the Nusselt number for a small element of the outside of a cylinder that has a height of $\Delta z$. I found the average Grashof number of a surface as $$Gr_{L}=\frac{\beta \rho ...
1
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0answers
108 views

Problem with cell size and boundary conditions in transient cylindrical conduction

I am attempting to model the steady state behavior of a cylinder using the finite volume method (FVM) subjected to a variety of boundary conditions in Matlab. First off, I am treating the cylinder as ...
5
votes
1answer
156 views

Interface Conductivity for Finite Volume Method Heat Transfer in Cylindrical Coordinates

I'm solving a heat conduction problem in cylindrical coordinates with a composite cylinder made of two different materials. Essentially the cylinder is split into a central cylinder of material A, ...
4
votes
1answer
121 views

Is is possible to mix finite-volume and finite-difference discretisations when solving a coupled non-linear system?

In my last question I noticed a peculiar error when solving the Poisson equation using finite volume method (FVM), Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) ...
6
votes
2answers
225 views

Is there a good tutorial or textbook-like source on implementing ENO/WENO with limiters in one (and more than one) dimension?

I've inherited a finite volume code that does a second-order discretization of flux terms for a set of mixed parabolic-elliptic equations with discontinuous diffusion coefficients. The impression I ...
4
votes
1answer
96 views

How can exponential fitting be used with the finite element method?

Restricted to one dimensional problem, is it possible to dynamically adapt the finite element method (FEM) discretisation based on the local value of the Péclet number ($P_e$) for advection-diffusion ...
4
votes
2answers
368 views

Does the finite element method impose any restrictions on the Peclet number for numerical stability?

Background on finite volume method When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...
6
votes
2answers
289 views

Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) finite volume method

I have been trying to debug this error the last few days I wondered if anybody has advice on how to proceed. I am solving the Poisson equation for a step charge distribution (a common problem in ...
6
votes
3answers
666 views

Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
4
votes
1answer
462 views

A simple function for generating a nonuniform mesh in 1D with fixed minimum spacing

I am solving an advection-diffusion problem where the solution variable is mostly flat apart from a small region near the centre of the domain where there are shape gradients. I would like to generate ...
25
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4answers
5k views

What are the conceptual differences between the finite element and finite volume method?

There is an obvious difference between finite difference and the finite volume method (moving from point definition of the equations to integral averages over cells). But I find FEM and FVM to be very ...
5
votes
2answers
210 views

Solving PDE with spatial and temporal derivatives on left hand side

I wish to solve an equation of the form, $$ \frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial x} \right) = -\frac{\partial}{\partial x}(\mathcal{F}) $$ for the variable $\phi$ (e.g. ...
1
vote
1answer
183 views

How to write this non-linear PDE with the finite volume method?

I wish to solve a coupled system of non-linear equation of this form, $$ u_t = -(\mathcal{F})_x + f(x,u,w) \\ w_t = \mathcal{F} + g(x,u,w) $$ by stepping the equations forward in time. The first ...
9
votes
2answers
319 views

Data structures for finite volume code: Arrays vs Classes

I have to write a finite volume code for Magnetohydrodynamics (MHD). I have written numerical code before but not at this scale. I just wanted to ask which will be a good choice, using a data ...
5
votes
2answers
366 views

Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?

I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" ...
2
votes
1answer
87 views

Finite volume with cell averages vs cell totals for conservation equations

What implementation details need to change if I use a cell average approach rather than a cell total approach for the finite-volume method? For example, consider the conservation law, $$ u_t + ...
13
votes
1answer
2k views

How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the ...
5
votes
1answer
236 views

Are we free to choose the position of ghost cells on a non-uniform finite-volume mesh?

Following Hundsdorfer approach the finite volume discretisation of the advection-diffusion equation (conservative form) on non-uniform cell centered grid can be written as, $$ w_j^{\prime} = ...
9
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3answers
302 views

How should non-constant coefficients be treated with finite-volume first order upwind scheme?

Starting with the advection equation in conservation form. $$ u_t = (a(x)u)_x $$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...
2
votes
1answer
50 views

Effect of Normalization in Unknowns

When solving a FV formulation of a set of equations, a code I am currently working with has user defined normalization factors for scaling equations. It normalizes time, number densities, potential, ...
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2answers
234 views

Connections between Differential Forms and the second order Finite Volume Method

Reading today about the theory of differential forms, I was left impressed how much it reminded me of second order Finite Volume Method (FVM). I'm struggling to figure out is thinking this way just ...
8
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1answer
1k views

OpenFoam vs FiPy

I need to learn and utilize a finite volume automated solution package for a project I'm working on and have narrowed it down to these two packages. I was wondering if anybody has experience of both ...
4
votes
1answer
775 views

How to approximate flux (with gradient) when using finite volumes?

In finite volume method one is using cell averages. In nonlinear conservation laws discontinuities can be created in the solution process. How to compute the flux when the flux contains a gradient ...